mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

Countdown probability, pt. 2

 2014-04-11 
As well as letters games, the contestants on Countdown also take part in numbers games. Six numbers are chosen from the large numbers (25,50,75,100) and small numbers (1-10, two cards for each number) and a total between 101 and 999 (inclusive) is chosen by CECIL. The contestants then use the six numbers, with multiplication, addition, subtraction and division, to get as close to the target number as possible.
The best way to win the numbers game is to get the target exactly. This got me wondering: is there a combination of numbers which allows you to get every total between 101 and 999? And which combination of large and small numbers should be picked to give the highest chance of being able to get the target?
To work this out, I got my computer to go through every possible combination of numbers, trying every combination of operations. (I had to leave this running overnight as there are a lot of combinations!)

Getting every total

There are 61 combinations of numbers which allow every total to be obtained. These include the following (click to see how each total can be made):
By contrast, the following combination allows no totals between 101 and 999 to be reached:
The number of attainable targets for each set of numbers can be found here.

Probability of being able to reach the target

Some combinations of numbers are more likely than others. For example, 1 2 25 50 75 100 is four times as likely as 1 1 25 50 75 100, as (ignoring re-orderings) in the first combination, there are two choices for the 1 tile and 2 tile, but in the second combination there is only one choice for each 1 tile. Different ordering of tiles can be ignored as each combination with the same number of large tiles will have the same number of orderings.
By taking into account the relative probability of each combination, the following probabilities can be found:
Number of large numbersProbability of being able to reach target
00.964463439
10.983830962
20.993277819
30.985770510
40.859709475
So, in order to maximise the probability of being able to reach the target, two large numbers should be chosen.
However, as this will mean that your opponent will also be able to reach the target, a better strategy might be to pick no large numbers or four large numbers and get closer to the target than your opponent, especially if you have practised pulling off answers like this.
Edit: Numbers corrected.
Edit: The code used to calculate the numbers in this post can now be found here.
                        
(Click on one of these icons to react to this blog post)

You might also enjoy...

Comments

Comments in green were written by me. Comments in blue were not written by me.
@Francis Galiegue: I've pushed a version of the code to https://github.com/mscroggs/countdown-...
Matthew
                 Reply
@Francis Galiegue: Sadly, I lost the code I used when I had laptop problems. However, I can remember what it did, so I shall recreate it and put it on GitHub.
Matthew
                 Reply
If you could, I'd love to have the code you used to do this exhaustive search?

I'm a fan of the game myself (but then I'm French, so to me it's the original, "Des chiffres et des lettres"), but for the numbers game, this is pretty much irrelevant to the language and country :)
Francis Galiegue
                 Reply
 Add a Comment 


I will only use your email address to reply to your comment (if a reply is needed).

Allowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li> <logo>
To prove you are not a spam bot, please type "graph" in the box below (case sensitive):

Archive

Show me a random blog post
 2025 

Jan 2025

Christmas (2024) is over
Friendly squares
 2024 
▼ show ▼
 2023 
▼ show ▼
 2022 
▼ show ▼
 2021 
▼ show ▼
 2020 
▼ show ▼
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

approximation national lottery folding paper talking maths in public a gamut of games royal institution exponential growth numbers bots frobel nine men's morris php propositional calculus electromagnetic field data 24 hour maths interpolation football live stream rhombicuboctahedron tennis wave scattering statistics estimation cross stitch probability flexagons hyperbolic surfaces recursion countdown newcastle triangles convergence big internet math-off gerry anderson fonts advent calendar finite group inverse matrices manchester pascal's triangle martin gardner kings asteroids pi approximation day pizza cutting london trigonometry curvature simultaneous equations python sound folding tube maps logo chebyshev preconditioning geometry dates draughts gaussian elimination databet platonic solids dinosaurs logs gather town go friendly squares pi rugby chalkdust magazine raspberry pi coins weak imposition oeis javascript stirling numbers programming european cup christmas card geogebra ternary fence posts reuleaux polygons computational complexity world cup matt parker sorting christmas numerical analysis royal baby sport signorini conditions mathsjam inline code squares manchester science festival matrices news accuracy errors chess wool cambridge youtube correlation tmip phd people maths data visualisation ucl mean stickers determinants realhats hexapawn matrix multiplication game of life runge's phenomenon books turtles mathsteroids bubble bobble games craft braiding menace crochet sobolev spaces reddit plastic ratio video games bodmas binary datasaurus dozen dragon curves guest posts harriss spiral finite element method map projections palindromes puzzles boundary element methods zines edinburgh error bars standard deviation arithmetic speed misleading statistics radio 4 logic dataset the aperiodical game show probability quadrilaterals matrix of minors machine learning golden ratio polynomials anscombe's quartet hats london underground captain scarlet hannah fry pac-man weather station mathslogicbot latex final fantasy golden spiral pythagoras graphs fractals graph theory crossnumber light bempp matrix of cofactors regular expressions noughts and crosses

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2025